COMPLETE INDEPENDENCE *I a. If a and b are in the class K, then a + 6 is in the class Km 'I b. // a and b are in the class K, then ab is in the class K. 4II a. There is an element Z such that a + Z = a for every element a. * 'II b. There is an element U such that aU = a for every element a. 'Ill a. a -f- b = 6 -f a. 'Ill b. ab = ba. •IV a. a -f be = (a -f 6) (a -f c). 'IV b. a(& + c) = ab + ac. 'V. Tor eeen/ element a there is an element a' such that a -f of = U and aa' — Z. 4VL There are at least too distinct elements in the class KS It will be readily seen that these postulates are satisfied by the following interpretation: a, b, c, ... are classes', a + b is the class of all those things that are in at least one of the classes, a, b; ab is the class of all those things that are in both of the classes a, b; Z is the 'null class' -the class that has no members; U is the 'universal class' - the class that contains all the things in all the classes under discussion. Postulate V then states that given any class a, there is a class a' consisting of all those things which are not in a. Note that VI implies that U, Z are not the same class. From such a simple and obvious set of statements it seems rather remarkable that the whole of classical logic can be built up symbolically by means of the easy algebra generated by the postulates. From these postulates a theory of what may be called 'logical equations' is developed: problems hi logic are translated into such equations, which are then 'solved' by the devices of the algebra; the solution is then reinterpreted in terms of the logical data, giving the solution of the original problem. We shall close this description with the symbolic equivalent of 'inclusion' - also interpretable, when propositions rather than classes are the elements of K9 as 'implication'. 'The relation a <b [read, a is included in b] is defined by any one of the following equations a + b = b, ab = a, a' + b = U, ab' = Z: 491